A wire has a resistance of 32 Q . It is melted and drawn into a wire of half of its original length. What is the resistance of the new wire?

1. Understanding Electric Current and Potential Difference (basic)

To understand electricity, we must first visualize what is happening inside a wire. Imagine a copper wire: it is filled with tiny particles called electrons. When these electrons start flowing in a specific direction, they create what we call an Electric Current. Think of it like water flowing through a pipe; the more water moving past a point every second, the stronger the current. In a circuit, this flow is actually a stream of electrons moving through a conductor Science, Class X (NCERT 2025 ed.), Chapter 11, p.192. However, there is a historical quirk you must remember: conventional current is always considered to flow from the positive terminal to the negative terminal, which is exactly opposite to the actual direction of electron flow.

But why do these electrons move at all? They need a "push." This push is provided by Potential Difference (often called Voltage). Imagine two water tanks connected by a pipe; if they are at the same level, no water flows. You need a height difference to create pressure. In electricity, a cell or a battery acts like a pump that creates this "electrical pressure" or potential difference across the ends of a conductor Science, Class X (NCERT 2025 ed.), Chapter 11, p.192. Without this difference in potential, electrons would simply wander aimlessly instead of forming a useful current.

To measure these quantities in a lab or a real-world circuit, we use specific instruments and units:

• Electric Current: Measured in Amperes (A). An ammeter is used for this and is always connected in series.
• Potential Difference: Measured in Volts (V). This is measured by a voltmeter, which must always be connected in parallel across the two points you are testing Science, Class X (NCERT 2025 ed.), Chapter 11, p.173.

Feature	Electric Current	Potential Difference
What is it?	The flow of electric charge (electrons).	The work done to move a charge between two points.
SI Unit	Ampere (A)	Volt (V)
Instrument	Ammeter (Series connection)	Voltmeter (Parallel connection)

Sources: Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.192; Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.173

2. Ohm's Law: The Relationship between V, I, and R (basic)

At the heart of understanding electricity lies Ohm’s Law, a fundamental principle discovered by George Simon Ohm. It states that the potential difference (V) across the ends of a metallic wire in a circuit is directly proportional to the current (I) flowing through it, provided its temperature remains constant Science, Chapter 11, p.176. Mathematically, this is expressed as V = IR.

The term Resistance (R) is the constant of proportionality for a given conductor. Think of resistance as the "electrical friction" that a material offers against the flow of charges. The SI unit of resistance is the ohm (Ω). By rearranging the formula to R = V / I, we can define 1 ohm as the resistance of a conductor when a potential difference of 1 Volt results in a current of 1 Ampere Science, Chapter 11, p.176.

Crucially, the resistance of a conductor is not just a random number; it depends on its physical dimensions and the nature of its material. Experimentally, it has been observed that the resistance (R) of a uniform metallic conductor is directly proportional to its length (l) and inversely proportional to its area of cross-section (A) Science, Chapter 11, p.192. This relationship is summarized by the formula:

R = ρ(l / A)

Where ρ (rho) is the electrical resistivity, a characteristic property of the material itself. This means if you make a wire longer, its resistance increases; if you make it thicker (increase area), its resistance decreases.

Factor	Change in Factor	Effect on Resistance (R)
Length (l)	Doubled	Doubles
Area (A)	Doubled	Halved
Temperature	Increased	Generally Increases (for metals)

Sources: Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.176; Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.192

3. Factors Affecting Resistance: Length and Area (intermediate)

To understand electrical resistance, think of it as the friction electrons encounter while moving through a conductor. This "struggle" is not random; it depends predictably on the physical dimensions of the wire. Scientific observation shows that the resistance (R) of a uniform metallic conductor is directly proportional to its length (l) and inversely proportional to its area of cross-section (A). Science, Class X (NCERT 2025 ed.), Chapter 11, p.178. Imagine a hallway: a longer hallway (length) is harder to navigate, while a wider hallway (cross-sectional area) allows more people to pass through comfortably.

These relationships are unified in the fundamental formula: R = ρ(l/A). Here, ρ (rho) is the constant of proportionality called resistivity. While resistance changes based on the shape of the object, resistivity is an intrinsic property of the material itself. For example, copper has a specific resistivity that doesn't change whether you have a thin wire or a thick block, but the resistance of that wire or block will differ based on its dimensions. Science, Class X (NCERT 2025 ed.), Chapter 11, p.179.

A sophisticated application of this concept occurs when a wire's shape is changed while its volume (V = A × l) remains constant—such as when a wire is melted and redrawn. If you modify the length, the area must change inversely to keep the volume the same. For instance, if a wire is redrawn to be half its original length (l/2), its thickness must double (2A) to account for the same amount of metal. Substituting these new values into our formula (R' = ρ[(l/2) / 2A]), we find the new resistance becomes one-fourth of the original. Science, Class X (NCERT 2025 ed.), Chapter 11, p.180.

Sources: Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.178; Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.179; Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.180

4. Electrical Resistivity (ρ) vs. Resistance (R) (intermediate)

To understand electricity, we must distinguish between an object’s dimensions and its inherent nature. Resistance (R) is a measure of how much an object opposes the flow of current. It is not just a property of the material, but also of its shape. As established in Science, Class X (NCERT 2025 ed.), Chapter 11, p.178, the resistance of a uniform metallic conductor is directly proportional to its length (l) and inversely proportional to its area of cross-section (A). Mathematically, this is expressed as:

R = ρ (l / A)

Here, ρ (rho) is the electrical resistivity. While resistance changes if you stretch or cut a wire, resistivity remains constant for a specific material at a given temperature. Think of it this way: Resistance is like the total travel time on a road (which depends on how long and wide the road is), while resistivity is like the quality of the pavement (an inherent characteristic of the material used). Metals and alloys have very low resistivity, making them excellent conductors, whereas insulators like glass have incredibly high resistivity Science, Class X (NCERT 2025 ed.), Chapter 11, p.179.

A critical concept for competitive exams is the conservation of volume during physical deformation. If a wire is melted and redrawn (reshaped), its volume (V = A × l) stays the same. Therefore, if you double the length (l), the cross-sectional area (A) must automatically become half to keep the volume constant. This dual change significantly impacts the resistance. For example, if a wire is compressed to half its length, its area doubles; substituting these into our formula (R = ρ [ (l/2) / 2A ]) shows the new resistance becomes one-fourth of the original.

Feature	Resistance (R)	Resistivity (ρ)
Definition	Opposition to current flow in a specific object.	Inherent property of the material itself.
Unit	Ohm (Ω)	Ohm-meter (Ω m)
Dependencies	Length, Area, Material, and Temperature.	Material and Temperature only.

Sources: Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.178; Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.179

5. Heating Effects and Power in Circuits (intermediate)

When an electric current flows through a conductor, it isn't a frictionless journey. Electrons constantly collide with the atoms and ions making up the conductor's lattice. Each collision transfers some kinetic energy from the moving electrons to the atoms, causing them to vibrate more vigorously. This internal microscopic agitation manifests macroscopically as heat. This phenomenon is known as the Heating Effect of Electric Current, an inevitable consequence of resistance in any real-world circuit Science, Class X (NCERT 2025 ed.), Chapter 11, p. 190.

To quantify this, we look to Joule’s Law of Heating. It states that the heat produced (H) in a resistor is directly proportional to the square of the current (I²), the resistance (R), and the time (t) for which the current flows. Mathematically, this is expressed as H = I²Rt. This relationship is vital for understanding why high-current appliances like geysers or air conditioners require thicker, heavy-duty wiring—even a small increase in current results in a squared increase in heat production Science, Class X (NCERT 2025 ed.), Chapter 11, p. 189.

Electric Power (P) is the rate at which this electrical energy is consumed or dissipated in a circuit. While the basic definition of power is work done per unit time, in electrical terms, it is the product of potential difference (V) and current (I). Depending on which variables you know, power can be calculated in three ways:

Formula	Best Usage Scenario
P = VI	When both voltage and current are directly measured.
P = I²R	Commonly used for series circuits where current remains constant.
P = V²/R	Commonly used for parallel circuits (like home wiring) where voltage is constant.

The SI unit of power is the Watt (W), representing 1 Joule of energy consumed per second Science, Class X (NCERT 2025 ed.), Chapter 11, p. 191. While heating is often seen as a "loss" (waste heat in computers), it is intentionally harnessed in devices like electric irons, toasters, and heaters. In an incandescent bulb, the filament is designed to retain enough heat to reach temperatures high enough to emit visible light Science, Class X (NCERT 2025 ed.), Chapter 11, p. 190.

Sources: Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.189; Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.190; Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.191

6. Combination of Resistors: Series and Parallel (intermediate)

To understand how circuits work, we must look at how resistors are combined. In a series combination, resistors are joined end-to-end so that there is only one path for the current to flow. Because there are no branches, the current (I) remains identical through every resistor in the series. However, the total potential difference (V) is divided among them. The equivalent resistance (Rₛ) is simply the sum of individual resistances: Rₛ = R₁ + R₂ + R₃... Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.192. You can visualize this as extending the length of a single wire; the longer the path, the higher the total resistance.

In contrast, a parallel combination connects resistors across the same two points. In this arrangement, the potential difference (V) is the same across every resistor, but the current divides into different branches. The rule here is different: the reciprocal of the equivalent resistance (1/Rₚ) is equal to the sum of the reciprocals of the individual resistances: 1/Rₚ = 1/R₁ + 1/R₂ + 1/R₃... Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.186. Interestingly, the total resistance in a parallel circuit is always less than the value of the smallest individual resistor because you are essentially providing more "lanes" for the electrons to travel through.

In domestic wiring, we almost exclusively use parallel circuits. Science, Class X (NCERT 2025 ed.), Chapter 12: Magnetic Effects of Electric Current, p.205. If appliances were in series, turning off one light would break the circuit for the entire house! Parallel connection ensures each appliance gets the full supply voltage and can be operated independently with its own switch.

Feature	Series Combination	Parallel Combination
Current (I)	Same through all resistors	Divides among branches
Voltage (V)	Divides across resistors	Same across all resistors
Total Resistance	Increases (Rₛ = R₁ + R₂...)	Decreases (1/Rₚ = 1/R₁ + 1/R₂...)

Sources: Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.192; Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.186; Science, Class X (NCERT 2025 ed.), Chapter 12: Magnetic Effects of Electric Current, p.205

7. Geometric Transformations and Constant Volume (exam-level)

To understand how the resistance of a wire changes when its shape is altered, we must look at the interplay between its physical dimensions and its material properties. The **Resistance (R)** of a conductor is defined by the formula **R = ρ(l/A)**, where 'l' is the length, 'A' is the cross-sectional area, and 'ρ' (rho) is the **resistivity**, a constant for a specific material at a given temperature Science, Class X, Electricity, p. 180. This tells us that resistance increases if the wire is made longer, but decreases if the wire is made thicker Science, Class X, Electricity, p. 178. However, a crucial concept in physics is the **Conservation of Volume**. When a wire is melted and redrawn or stretched, the total amount of material — its volume (V) — remains unchanged. Since the volume of a cylindrical wire is the product of its cross-sectional area and its length (**V = A × l**), any change in length must be compensated by a proportional change in the area. For instance, if you compress a wire to make it shorter, it must become thicker to accommodate the same volume of metal. Let’s look at the mathematical impact of this transformation. If a wire is reshaped such that its new length (l') is exactly half the original length (l/2), the area (A') must double (2A) to keep the volume constant (V = l/2 × 2A = lA). When we plug these new dimensions into our resistance formula, we find that the new resistance (R') is:
R' = ρ(l/2) / (2A) = 1/4 × ρ(l/A) = R/4.
In this scenario, the resistance doesn't just halve; it drops to one-fourth of its original value because both the reduced length and the increased thickness work together to make it easier for current to flow.

Geometric Change (Constant Volume)	Effect on Area	Effect on Resistance
Length is Doubled (2l)	Area is Halved (A/2)	Increases by 4 times (4R)
Length is Halved (l/2)	Area is Doubled (2A)	Decreases to 1/4th (R/4)

Sources: Science, Class X, Chapter 11: Electricity, p.178; Science, Class X, Chapter 11: Electricity, p.180

8. Solving the Original PYQ (exam-level)

Now that you have mastered the fundamental formula for resistance, it is time to see how UPSC tests your ability to apply these building blocks simultaneously. This question hinges on the Law of Conservation of Volume. As you learned in Science, class X (NCERT 2025 ed.) > Chapter 11: Electricity, when a wire is melted and reshaped, its volume remains constant even as its dimensions change. The core relationship here is R = ρ(l/A). If the length is compressed, that material doesn't disappear; it must go somewhere, meaning the wire becomes thicker as it gets shorter.

Let’s walk through the coach's logic: if the new length (l') is half the original (l/2), the cross-sectional area (A') must double (2A) to keep the total volume (V = A × l) the same. When you substitute these values into the resistance formula, the change is compounded: the halved length reduces resistance by half, and the doubled area reduces it by half again. This results in a new resistance that is exactly 1/4th of the original value. Therefore, 32 Ω divided by 4 gives us the correct answer, (C) 8 Ω.

In the examination hall, UPSC expects you to avoid the common "linear thinking" traps. Option (B) 16 Ω is the most frequent mistake; it attracts students who only account for the change in length but forget the simultaneous change in area. Option (A) 32 Ω wrongly assumes resistance is an intrinsic property like resistivity (ρ) that doesn't change with shape. By recognizing that resistance depends on the geometry of the conductor, you move beyond rote memorization to true conceptual clarity.